PHYSICAL REVIEW A 93, 042314 (2016)

Macroscopic entanglement in many-particle quantum states

Malte C. Tichy,1 Chae-Yeun Park,2 Minsu Kang,2 Hyunseok Jeong,2 and Klaus Mølmer1

1Department of Physics and Astronomy, University of Aarhus, DK-8000 Aarhus C, Denmark2Center for Macroscopic Quantum Control, Department of Physics and Astronomy, Seoul National University, Seoul 151-742, Korea

(Received 29 July 2015; revised manuscript received 30 December 2015; published 11 April 2016)

We elucidate the relationship between Schrodinger-cat-like macroscopicity and geometric entanglement andargue that these quantities are not interchangeable. While both properties are lost due to decoherence, we showthat macroscopicity is rare in uniform and in so-called random physical ensembles of pure quantum states,despite possibly large geometric entanglement. In contrast, permutation-symmetric pure states feature rather lowgeometric entanglement and strong and robust macroscopicity.

DOI: 10.1103/PhysRevA.93.042314

I. INTRODUCTION

Quantum entanglement entails two important conse-quences: On the one hand, it is the “characteristic trait ofquantum mechanics” [1] that thoroughly thwarts our everyday’s intuition, most bizarrely when applied to macroscopicobjects, as illustrated by the famous paradox of Schrodinger’scat [2]. On the other hand, entanglement is the very ingredientthat makes the simulation of quantum many-body systemsextremely challenging: A separable system of N qubitsrequires only 2N parameters for its description, whereas anentangled state comes with ∼2N variables. This curse ofdimensionality in the context of simulating quantum systemsbecomes a powerful resource when it comes to the speedup ofquantum computers over classical architectures. Although thetwo aspects of entanglement are two sides of the same medal,they are quantified differently: Measures of macroscopicity [3]reflect the degree to which a quantum state resembles aSchrodinger’s cat; the complexity of a quantum state isreproduced by the geometric measure of entanglement [4],defined via the largest overlap to separable states. Macro-scopicity can increase under local operations combined withclassical communication (LOCC) [3] in contrast to geometricentanglement, e.g., the modestly macroscopic cluster statecan be converted via LOCC into a highly macroscopic GHZstate [5].

In nature, macroscopic entanglement does not occur [6,7]outside artificially tailored situations [8]. Its empirical absencecontrasts with its immediate appearance in the formalism ofquantum physics, which raises the question whether some un-avoidable mechanism jeopardizes macroscopically entangledquantum states. The decoherence program [9] explains theemergence of classical behavior in our everyday world and howmacroscopic quantum superpositions decohere on overwhelm-ingly short time scales [10–13]: The interaction between anyquantum system and its surrounding environment destroysthe coherence between macroscopically distinct alternatives—such as the dead and the living cat. Even if the environmentwere shielded off perfectly, however, it remains unlikely thatmacroscopic entanglement be observed in nature, due to theunavoidable coarse graining of any measurement [14–16].In other words, there are powerful mechanisms that quicklydeteriorate macroscopic entanglement.

But the decoherence program does not make any statementon the likelihood that a macroscopic quantum superposition

may form, only that, when it appears, it decoheres on atime scale so short that any attempt for observation isvain. Would a hypothetical decoherence-free world then hostcohorts of Schrodinger’s cats? In other words, how likely aremacroscopically entangled quantum states before the onset ofdecoherence?

In order to ease our intuition for these questions, let uspropose a classical analogy, which is meant here in a qual-itative and metaphorical sense: Consider the microcanonicalensemble of a gas of N particles in a box with total energyE, illustrated in Fig. 1. In classical statistical mechanics, allmicrostates (specified by the positions and momenta of the gasparticles {�x1, . . . ,�xN ; �p1, . . . , �pN }) that are compatible withthe total energy E are assigned the same probability [17].How likely is it to find all particles in the left half of the box,as illustrated in Fig. 1(b)? In the first place, we can adopt adynamical explanation: If we prepared the particles in sucha state, the system would relax to a homogenous distribution[Fig. 1(a)] on a very short time scale—just like decoherencedynamics destroy any macroscopic quantum superposition.

This explanation for the absence of the strongly inhomoge-neous situation is, however, not the commonly adopted one:The spontaneous occurrence of the macrostate sketched inFig. 1(b) is by itself extremely unlikely, and the relaxationdescribed above is in fact not required to explain its rarity.There are overwhelmingly more microstates that correspondto a homogeneous distribution (a) than for a distribution withall particles on one side (b) [17]. That is, it is eventuallya statistical argument that allows us to safely neglect theinhomogeneous macrostate and focus on the macrostate with ahomogeneous distribution of particles—which is the very basisof statistical mechanics [17]. In other words, even though thereis a powerful dynamical mechanism to restore the homogeneityof the gas, this mechanism is not required to explain theabsence of inhomogenous distributions: These macrostates aresufficiently unlikely to occur to be safely neglected.

Quite similarly, anticipating our results below, despitedecoherence being a powerful dynamical mechanism toexplain the disappearance of macroscopic quantum super-positions [10,12], these states are a priori extremely rarein many ensembles of pure states. As a consequence, theabsence of macroscopic superpositions can be understoodfrom a statistical argument.

While mixed states already include the effect of deco-herence and feature little macroscopicity [3], we focus here

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TICHY, PARK, KANG, JEONG, AND MØLMER PHYSICAL REVIEW A 93, 042314 (2016)

(a) (b)

FIG. 1. Two microstates of the microcanonical ensemble of 2000particles in a box. (a) Typical states feature a rather homogeneousdistribution of particles, while (b) states with large inhomogeneitiesare artificial and rare. We argue in this article that the rarity ofmacroscopically entangled pure quantum states can be understood inclose analogy: Macroscopically entangled states play the role of stateswith large inhomogeneities (b), and therefore form only extremelyseldom.

on pure states. The typicality of macroscopic entanglementthen depends on the actual choice of the pure-state ensemble.We find analytical and numerical evidence that macroscopicsuperpositions are untypical among random pure states indifferent ensembles. This finding may appear paradoxicalat first sight, since macroscopicity leads to multipartiteentanglement [18] and entanglement, as quantified by thegeometric measure [4,19] (tantamount to a large distance toseparable states), is common in random pure states [20,21]and macroscopicity is bound from above by a monotonicallyincreasing function of geometric entanglement [22]. Weresolve this ostensible paradox by showing that geometricentanglement is actually adverse for macroscopicity: Randomstates are very entangled, and hence nonmacroscopic.

We review measures of macroscopic and geometric en-tanglement in Sec. II, where we also present some technicalresults. Since both quantities are defined via a maximizationprocedure, their relationship is intricate. We elucidate theirconnection qualitatively in Sec. III, in order to gain a goodunderstanding of the statistics of macroscopic entanglementevaluated in different pure-state ensembles in Sec. IV. Weconclude in Sec. V, where we propose an extension of ourstudy to other ensembles and sketch its consequences for thepreparation of macroscopic states in the experiment.

II. QUANTIFYING MACROSCOPIC AND GEOMETRICENTANGLEMENT

In order to address our central question—whether macro-scopicity is typical in ensembles of pure quantum states—weneed to establish a quantitative measure for macroscopicity.Our system of interest is a collection of N qubits, which,for the purpose of illustration, we treat as spin 1/2 particles.That is, we focus on pure quantum states living in the Hilbertspace H = (C2)

N. While no consensus exists on how to

rigorously quantify macroscopicity for mixed states [3,23–28],this debate is not crucial in our context, since we focus on purestates, for which most measures agree. We will adapt a well-established measure for macroscopic entanglement [3,29] andpropose a sensible way for its normalization. To set the

context, we are interested in the typicality of macroscopicentanglement within quantum theory; a general benchmarkof macroscopic quantum superpositions will require conceptsthat draw beyond this realm [30,31], and may be tailored forspecific applications [32].

A. Measure of mascroscopicity

Macroscopicity manifests itself in disproportionally largefluctuations of some additive multiparticle observable [3,33],i.e., of some operator of the form,

S(�α1, . . . �αN ) =N∑

j=1

�αj · �σj , (1)

where �σj is the vector of three Pauli matrices that act on the j thqubit and the local orientation of the measurement operator �αj

is normalized,

|�αj |2 = 1. (2)

The operator S describes the total spin of the system withrespect to locally adjusted spin directions, defined by �αj =(αx

j ,αy

j ,αzj ). Given a pure quantum state |�〉, the maximally

obtainable variance of this additive operator,

〈�S2(�α1, . . . �αN )〉 = 〈�|S2|�〉 − (〈�|S|�〉)2, (3)

defines the unnormalized macroscopicity M [3],

M = max�α1,...,�αN〈�S2(�α1, . . . �αN )〉. (4)

Quite naturally, we find

N � M � N2. (5)

The lower bound is saturated, e.g., for separable states, forwhich we maximize fluctuations by choosing the measure-ment directions �αj to be unbiased with respect to the localspin directions. The upper bound is reached, e.g., for theGreenberger-Horne-Zeilinger (GHZ) state [34],

|GHZ〉N = 1√2

(|0〉⊗N + |1〉⊗N ), (6)

which comes closest to modeling a coherent superposition ofa living and a dead cat: The two superimposed alternatives(all spins pointing up, all spins pointing down) are maximallydifferent, leading to maximal fluctuations.

In order to compare the macroscopicity of systems ofdifferent sizes N , we normalize M:

M(|�〉) =√M(|�〉) − N

N (N − 1), (7)

such that 0 � M � 1 for all N , where the upper (lower) boundis saturated for M = N2 (N ).

Strictly speaking, the normalized macroscopicity M is nota degree of quantum macroscopicity such as those studiedin Refs. [3,23–28]. It should be understood as a degree ofmacroscopic quantum coherence normalized by the size of thephysical system under consideration.

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B. Additivity and basic properties

The unnormalized macroscopicity M is additive in thesense that any product state |�〉 ⊗ |�〉 yields

M(|�〉 ⊗ |�〉) = M(|�〉) + M(|�〉), (8)

because the separability of |�〉 ⊗ |�〉 excludes additionalfluctuations by choosing a direction of spins other than theoptimal ones for |�〉 and |�〉.

Any family of states for which the size of nonseparablecomponents does not scale with the system size has vanishingnormalized macroscopicity in the limit of many particles. Forexample, a tensor product of N/2 Bell states, a two-produciblestate [35], is not macroscopic:

M(|�−〉⊗N/2) = 2N, (9)

M(|�−〉⊗N/2) = 1√N − 1

N→∞→ 0, (10)

which agrees with the upper bound on the quantum Fisherinformation found in Ref. [36].

C. Relation to index p

Our measure of macroscopicity can be directly related tothe index p [37], the exponent that defines the scaling of Mwith N ,

M ∝ Np. (11)

The scaling properties of macroscopic entanglement can beinvestigated for state families, i.e., “prescriptions that assign toany system size a quantum state |�〉” [33]. Any state family forwhich M > δ > 0 with δ independent of N can be consideredmacroscopic: In the limit N → ∞, M is then related to theindex p as follows:

M = 0 ⇔ p = 1, (12)

M > δ > 0 ⇔ p = 2. (13)

In other words, for p = 2, we have macroscopically largefluctuations that do not vanish in the limit of many particles;the index p, however, does not yield any information aboutthe actual fraction of particles participating in a macroscopicsuperposition. The normalized macroscopicity M is more finegrained and offers such insight: For the family of states,∣∣�(N1,N2)

⟩ = |GHZ〉N1 ⊗ |1〉⊗N2 , (14)

we have

M(|�〉) = N21 + N2, (15)

and, thus,

M(|�〉) N1,N2�1≈ N1

N1 + N2. (16)

In this case M reflects the fraction of particles in the systemthat takes part in a quantum superposition of macroscopicallydistinct alternatives. The proportionality of M with the size ofthe macroscopic subsystem represents our motivation to applya square root in the definition ofM(|�〉) in Eq. (7), besides Mbeing of square order due to its relation to the variance (1). On

the other hand, for large N , we also have NM ≈√M (with an

additive error of the order ∼√N ), which quantifies the absolute

size of the macroscopic superposition. In the following, wewill therefore focus on the macroscopicity M, as defined inEq. (7).

D. Evaluation of macroscopicity

1. Generic states

The definition of the unnormalized macroscopicity M inEq. (4) entails an optimization problem over 2N variables,which evidently complicates its evaluation for large systems.As an alternative to multivariable optimization, it was pro-posed [38] to evaluate the unnormalized macrocopicity Musing the variance-covariance matrix,

Vγk,βj = 〈�|�σγ

k �σβ

j |�〉, (17)

where �σγ

l = σγ

l − 〈σ γ

l 〉, γ = x,y,z. The resulting 3N × 3N

matrix V then stores the fluctuations of all observables that aresums of Pauli matrices, and we have [38]

〈�S2(�α1, . . . ,�αN )〉 =N∑

j,k=1

∑γ,β=x,y,z

αγ

j Vαj,βkαβ

k . (18)

The expectation value 〈�S2(�α1, . . . ,�αN )〉 is maximized bychoosing the �αj as the eigenvector �v1 of V corresponding tothe largest eigenvalue λ1 of V , such that

�αj =√

N

⎛⎜⎝

v1,3(j−1)+1

v1,3(j−1)+2

v1,3(j−1)+3

⎞⎟⎠. (19)

However, the �αj chosen this way are only constrained by

N∑l=1

|�αl|2 = N, (20)

which is a much weaker constraint than our Eq. (2): Instead ofN unit-normalized Bloch vectors �αj , only the sum of the normof all Bloch vectors is fixed in Eq. (20). Colloquially speaking,the �αj chosen according to Eq. (19) [38] allow us to weightthe importance of individual spins in the system differently,and give those featuring large fluctuations a larger impact. Wecan therefore only state that

M(|�〉) � Nλ1. (21)

Admittedly, the scaling of λ1 with the system size N for agiven family of states is inherited by M evaluated via Eq. (4),such that coarse-grained quantities such as the index p can beevaluated using λ1. In particular, the eigenvalues of V only takethe values 1 and 0 for separable states, which are clearly—andnot surprisingly—not macroscopic at all. For a quantitativeunderstanding of macroscopicity, however, the optimizationinherent to (4) is crucial: For example, consider the state,

|�c〉 = |�+〉 ⊗ |0〉⊗N−2, (22)

with N � 3, where the first two qubits are in a maximallyentangled Bell state, but remain completely separable fromthe rest of the system. The unnormalized macroscopicity of

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|�c〉 fulfills

M = N + 2 < 2N = λ1N, (23)

i.e., the largest eigenvalue λ1 is related to the eigenvector �v1,with

�α1 =√

N

2(1,0,0), (24)

�α2 =√

N

2(1,0,0), (25)

�αk�3 = (0,0,0), (26)

which is not compatible with Eq. (2) and entirely ignores theseparable qubits 3, . . . ,N , while the two entangled qubits areover-weighted. This example being admittedly artificial, wehave nevertheless experienced a substantial difference betweenthe exact calculation and the value extracted via the VCMfor the nonsymmetric ensembles of random states consideredbelow in Sec. IV.

On the other hand, given a set of orientations obtainedvia (19) and |�αj |2 > 0 for all j , we can normalize the �αj tofind a candidate spin orientation that promises to yield a largevariance,

�βj = �αj√|�αj |2. (27)

The resulting value 〈�S2( �β1, . . . , �βN )〉 then provides a lowerbound on the actual unnormalized macroscopicity, since weare not guaranteed that the choice of local spin orientationsgiven by (27) is the optimal one:

〈�S2( �β1, . . . , �βN )〉 � M. (28)

In other words, even though the precise value of M requiresa numerical optimization, computationally inexpensive lowerand upper bounds [Eqs. (28) and (21), respectively] to thisquantity can be established straightforwardly.

2. Symmetric states

In the case of permutation-symmetric states, V assumesa structure with repeated 3 × 3 blocks, and the eigenvectors�vj reflect this symmetry. As a consequence, the optimallocal spin orientations all coincide, �αl = �αk for all k,l, and,consequently, �βj = �αj . The optimization inherent to Eq. (4)becomes unnecessary, since the lower bound (28) and the upperbound (21) on the unnormalized macroscopicity coincide. Wecan therefore safely adopt the method introduced in Ref. [38]to compute M.

Specifically, the variance-covariance matrix V then consistsof two different 3 × 3 blocks,

Aγ,β = Vγ 1,β1, (29)

Bγ,β = Vγ 1,β2, (30)

which contain all variances and covariances, respectively, andassumes the structure,

V =

⎛⎜⎜⎝

A B . . . B

B A . . . B...

.... . .

...B B . . . A

⎞⎟⎟⎠. (31)

By writing V = 1 ⊗ A + M ⊗ B, one can show that the largesteigenvalue λ1 of the above block matrix V coincides with thelargest eigenvalue of the 3 × 3 matrix,

Vsym = A + (N − 1)B, (32)

such that

M(|�sym〉) = Nλ1. (33)

This greatly facilitates the computation of the macroscopicityfor permutation-symmetric states. To obtain the matrices A

and B, we use the efficient techniques for the computationof reduced density matrices of symmetric states presented inRef. [39].

E. Geometric measure of entanglement

We will relate macroscopicity to the geometric measure ofentanglement [4,40], which is defined via the maximal overlapη of |�〉 with any separable state |�sep〉 = |φ1,φ2, . . . ,φN 〉,

EG(|�〉) ≡ − log2 η = − log2 sup|�sep〉

|〈�sep|�〉|2. (34)

The geometric measure of entanglement of N qubits naturallyvanishes for separable states, and is bounded from above byN − 1. High geometric entanglement is tantamount to a largegeneralized Schmidt measure [41], which reflects the numberof separable terms required to express the state; the geometricmeasure of entanglement, thus, is reciprocal to the complexityof the quantum state, and thereby represents the “other side ofthe medal” of quantum entanglement.

Since we will face the statistics of geometric entanglementin Sec. IV, we discuss its evaluation in practice. In general,the computation of the geometric measure of entanglement re-quires an optimization over the 2N free parameters x1, . . . ,xN

and y1, . . . ,yN that define the separable state,

|�sep〉 = ⊗Nj=1(cos xj |0〉 + eiyj sin xj |1〉), (35)

entailing significant computational expenses. Alternatively,candidate solutions for the closest separable state can becomputed via the probabilistic iterative algorithm presentedin Ref. [19].

For permutation-symmetric states, the evaluation of thegeometric measure is facilitated considerably. A permutation-symmetric state of N qubits can be written in the Majoranarepresentation,

|�sym〉 = 1√N

∑σ∈SN

⊗Nj=1

∣∣εσj

⟩, (36)

where

N = N !perm(〈εj |εk〉) (37)

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is a normalization constant, and perm(〈εj |εk〉) is the perma-nent [42] of the N × N Gram matrix that contains all mutualscalar products 〈εj |εk〉. The permanent is, in general, a functionthat is exponentially hard in the matrix size N , but for a Grammatrix with only two nonvanishing singular values, as givenhere by construction, an efficient evaluation is possible [42].For this purpose, the representation of symmetric states in theDicke basis is valuable,

|�sym〉 =N∑

j=0

cj

∣∣D(j )N

⟩. (38)

The Dicke states are defined as

∣∣D(j )N

⟩ =(

N

j

)−1/2 ∑σ∈S{1,...,1,0,...,0}

⊗Nj=1|σj 〉, (39)

where the summation includes all possibilities to distributej particles in |1〉 and N − j particles in |0〉 among the N

modes. The quantitative relationship between the expansioncoefficients in the Dicke basis cj and the N states |εk〉that define the Majorana representation (36) is presented inRef. [43].

Since the closest separable state to a symmetric state is itselfsymmetric [44], the optimization problem over 2N variablesimplicit in Eq. (34) reduces to a merely two-dimensionalsetting. Given the Majorana representation |ε1〉, . . . ,|εN 〉, weneed to find the single-qubit state |φ〉 that maximizes

1

N

N∏j=1

|〈εj |φ〉|2. (40)

Besides standard numerical optimization strategies, we canadapt the iterative algorithm of Ref. [19] to symmetric states:For that purpose, we choose a random single-particle state|φ0〉. We then iteratively generate states |φk〉 as follows:

|φk+1〉 =N∑

j=1

|εj 〉〈εj |φk〉 , (41)

|φk+1〉 = |φk+1〉√〈φk+1|φk+1〉

. (42)

The results of Ref. [19] can be translated to a wide extent tothe current setting with symmetric states, and |φk〉 becomesa good candidate for the closest separable state for large k,although the algorithm is prone to return a local instead of aglobal maximum.

III. RELATIONSHIP BETWEEN MACROSCOPICITY ANDGEOMETRIC ENTANGLEMENT

Having established the two pertinent main characteristicsof a quantum many-body state |�〉—its macroscopicity M[Eq. (7)] and its geometric entanglement EG [Eq. (34)]—wecan now explore the relationship between these two quantities.

The behavior of entanglement measures and macroscopic-ity was analyzed in Ref. [18], where the author finds that “astate which includes superposition of macroscopically distinctstates also has large multipartite entanglement in terms ofthe distance-like measures of entanglement.” Here, we will

confirm that geometric entanglement is indeed necessary fornonvanishing macroscopicity; however, we will also show thatthe general relationship between macroscopicity and geomet-ric entanglement is rather involved. In particular, nonvanishinggeometric entanglement is not sufficient for macroscopicity:There are entangled states with strictly vanishing macroscop-icity. On the other hand, very large geometric entanglementimplies small macroscopicity, i.e., the maximal value ofmacroscopicity is reached for finite geometric entanglement.As a consequence, the two quantities should not be usedsynonymously.

A. Close-to-separable states

We first focus on states |�〉 for which the largest squaredoverlap with a separable state η is larger than or equal to 1/2,i.e., the geometric measure of entanglement EG(|�〉) [Eq. (34)]is smaller than or equal to unity.

To this end, we explore the transition between the separablestate |0, . . . ,0〉, which carries neither macroscopicity norentanglement, to the maximally macroscopic GHZ state (6),described by

|�(θ,ε)〉 = cos θ |0〉⊗N + sin θ (cos ε|0〉 + sin ε|1〉)⊗N√1 + cosN ε sin(2θ )

,

(43)

with 0 � ε � π/2 and 0 � θ � π/4. The state |�(θ,ε)〉 is asuperposition of two separable components in which all qubitspopulate the very same states, with weights depending on θ .The distinguishability of the two alternatives is defined by ε.The parametrization in ε for fixed θ = π/4 was introducedin Ref. [23] and explored in Refs. [3,45]. Since the state ispermutation symmetric, we can use the methods of Sec. II D 2to evaluate its macroscopicity.

Several limiting cases are reached for particular values ofthe parameters θ,ε: For (θ,ε) = (π/4,π/2), we recover theGHZ state (6); for θ = 0 or ε = 0, we deal with a separablestate. For ε > π/2, the destructive interference between thetwo amplitudes associated with the component |0〉⊗N can leadto geometric entanglement larger than unity. In particular, weobtain the W state,

|W 〉 ≡ ∣∣D(1)N

⟩ = 1√N

(|1,0, . . . ,0〉 + |0,1,0, . . . ,0〉 . . . ),

(44)

in the limit ε → π,θ → π/4, for odd N .We parametrize the closest separable state as

|�sep(α)〉 = (cos α|0〉 + sin α|1〉)⊗N, (45)

the overlap with |�(θ,ε)〉 becomes

|〈�sep(α)|�(θ,ε)〉|2 = (cos θ cosN α + sin θ cosN (ε − α))2

1 + cosN ε sin(2θ ),

(46)

which needs to be maximized with respect to α to obtain thegeometric measure of entanglement. For large N , the overlapis maximized for α = 0 (since θ � π/4); for finite N , themaximum can conveniently be found numerically, since theoverlap (46) does not oscillate fast as a function of α.

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FIG. 2. Macroscopicity as a function of geometric entanglementfor the family of states |�(θ,ε)〉. The two extremal cases are given byε = π/2 (black dashed) and θ = π/4 (solid red), which take turns asthe upper bound on macroscopicity for given geometric entanglement.The blue line is the upper bound of Eq. (47).

We show the behavior of geometric and macroscopicentanglement in Fig. 2, for different numbers of qubits N .Although macroscopicity increases with geometric entangle-ment as a general tendency, the relationship is ambiguous,especially for large numbers of qubits N . Based on extensivenumerical evidence, we conjecture that the general maximummacroscopicity for a given value of geometric entanglementis attained by the value obtained for |�(θ = π/4,ε)〉 or|�(θ,ε = π/2)〉, in the range EG � 1. We could not find aproof for this conjecture, which implies a tighter bound on Mthan the inequality,

M �√

6√√

(1 − 2−EG )N√

N − 1, (47)

which we adapted from Eq. (8) of Ref. [22], shown in Fig. 2as a blue line.

1. Geometric entanglement without macroscopicity

We first consider the family of states parametrized by ε =π/2,0 � θ � π/4 (black dashed lines in Fig. 2); the pertinentvariance-covariance matrix Vsym in Eq. (32) then becomes

Vsym =

⎛⎜⎝

1 0 0

0 1 0

0 0 N sin2 2θ

⎞⎟⎠, (48)

which yields the macroscopicity,

M =√

max(1,N sin2 2θ ) − 1

N − 1. (49)

The geometric entanglement takes the value

EG = − log2(cos2 θ ). (50)

As a consequence, for 0 < θ � 12 arcsin

√1/N , the geometric

entanglement remains finite, yet the macroscopicity vanishes.In other words, there are states that are entangled, but thefluctuations in any additive observable do not surpass thosethat can be achieved for separable states. This explains the

steplike behavior observed in Fig. 2, best visible for smallN . However, the range of EG > 0 for which M = 0 shrinkswith increasing N , which poses the general question of whichis the lower bound to M in terms of EG, i.e., whether athreshold �(N ) can be found such that EG > �(N ) impliesnonvanishing macroscopicity.

On the other hand, even though geometric entanglementis not sufficient for macroscopicity, it is necessary: For aseparable state, all nonvanishing eigenvalues of the variance-covariance matrix (17) are unity.

2. Close-to-unity geometric entanglement and smallmacroscopicity

The most macroscopic state, the GHZ state (6), pos-sesses geometric entanglement EG(|�GHZ〉) = 1. AnticipatingSec. III B, close-to-maximal macroscopicity (very close tounity) naturally comes with geometric entanglement EG ≈ 1,in general. The criterion EG ≈ 1 is, however, not sufficient toensure large macroscopicity: Consider the family of statesparametrized by θ = π/4,0 � ε � π/2. By evaluating thelargest eigenvalue of the matrix Vsym, we obtain the macro-scopicity,

M =√

sin2 ε

1 + cosN ε. (51)

The maximal overlap to separable states is bounded frombelow by the overlap with the separable test-state (45) settingα = 0; the geometric measure of entanglement thereforefulfills

EG(|�(θ = π/4,ε)〉) � − log2(1 + cosN ε)

2. (52)

That is, for a wide range of ε � π/2, we retain a geometricmeasure of entanglement of around unity, but quickly losemacroscopicity. In Fig. 2, the red lines quickly dive into lowvalues of macroscopicity, while remaining close to EG = 1, atrend that becomes more and more clear for larger values of N .This stands in stark contrast to the black dashed lines, whichretain macroscopicity for decaying geometric entanglement.

This behavior can be understood intuitively via the de-composition of the state into Dicke states [Eq. (38)], shownin Fig. 3. The largest overlap with any separable state is atleast as large as the coefficient in the Dicke-state expansionrelated to |D(0)

N 〉, since the latter is separable. For decreasingε ≈ π/2, we continuously lose macroscopicity, because theaverage directions of the spins become similar and thetwo superimposed alternatives less and less macroscopicallydistinct. However, the closest separable state remains theDicke state |D(0)

N 〉 for a wide range of ε � π/4, i.e., thegeometric measure of entanglement remains close to unity.In contrast, for the family parametrized by ε = π/2, the lossof geometric entanglement is directly accompanied by a lossof macroscopicity (upper panel of Fig. 3).

B. Far-from-separable states

Let us now move into the domain of strongly geometricallyentangled states and assume that we are given a maximaloverlap with separable states η < 1/2, i.e., a geometric

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FIG. 3. Decomposition of |�(θ,ε)〉 for N = 24 into Dicke-statecomponents [Eq. (39)], for the two extremal families of statescharacterized by ε = π/2 (upper panels) and θ = π/4 (lower panels).For ε = π/2, only the very first and very last Dicke componentsare populated, while a binomial distribution of components slowlyshifts to the highest Dicke component for θ = π/4. Consequently,the behavior in the EG-M plane (Fig. 2) is very different for thetwo families of states. In the lower right panel, the black dashed lineshows the upper bound to EG given in Eq. (52).

measure of entanglement EG = − log2 η > 1. We constructa state |�η〉 that maximizes the macroscopicity under thisconstraint. Since local rotations do not affect the geometricmeasure of entanglement, we can assume that the optimalvalues of the spin orientations all point into the z direction(αj = (0,0,1)), and we expand |�〉 in eigenstates of the totalspin operator S, which has eigenvalues −N,−N + 2, . . . ,N .Each eigenvalue is

(N

(N + S)/2

)-fold degenerate; the state can

therefore be written as

|�η〉 =∑

S=−N,−N+2,...N

(N

(S + N)/2

)∑λ=1

cS,λ|S,λ〉, (53)

where λ labels the degenerate states, and we choose the |S,λ〉 tobe separable. The expectation value of powers of the collectivespin becomes

〈�|Sk|�〉 =∑

S=−N,−N+2,...N

Sk

(N

(S + N)/2

)∑λ=1

|cS,λ|2. (54)

Under the constraint that the maximal overlap to any separablestate be fixed to η,

|cS,λ|2 � η, (55)

we maximize the variance (3) by setting cN,1 = c−N,1 =√η and subsequently distributing the probability amplitude√1 − 2η among the remaining coefficients, i.e., we set as

many pairs of coefficients to cS,k = c−S,k = √η as possible,

proceeding from large to small total spins S. Effectively, weset � 1

η� coefficients to the maximal overlap

√η, choosing these

pairwisely to yield as large as possible a contribution to 〈S2〉without any contribution to 〈S〉, and a last pair of coefficients

to√

1 − η� 1η�/√2. This way, we maximize the contribution

to the expectation value of S2, while the expectation value ofS remains 0. The last pair cS,k = c−S,k is set to accommodatethe remaining amplitude, typically smaller than

√η. Formally,

the state reads

|�η,max〉 =∑

S=−N,−N+2,...,−mod(N,2)

×(

N

(N − S)/2)∑k=1

√η(eiφS,k |S,k〉 + eiφ−S,k | − S,k〉), (56)

where the sum only runs over so many terms such that the stateis normalized to unity; one term may possibly be weighted bya factor smaller than

√η.

The obtained bound is shown in Fig. 4 for different values ofN , as a function of geometric entanglement. The maximallyachievable macroscopicity grows as a function of N for afixed overlap η, but, for a fixed number of qubits N , asmall overlap η, equivalent to large geometric entanglement,causes a reduced macroscopicity. While we have normalizedthe macroscopicity to 0 � M � 1, allowing a comparisonof different system sizes, the maximal value of geometricentanglement N − 1 depends on N ; hence a normalizedversion of the geometric entanglement (lower panel of Fig. 4)exhibits a more homogeneous decay. The expansion intoseparable states Eq. (53) is, however, not necessarily theoptimal generalized Schmidt decomposition [41], i.e., it isoften possible to find a separable state with overlap largerthan η. We can therefore not expect the bounds to betight.

We can repeat the argument for symmetric states, for whichwe impose that the amplitude of each Dicke-state component|D(k)

N 〉 is constrained by η, leading to a state of the form,

|�η,max〉 =∑

k=0...�N/2�

√η[eiφk

∣∣D(k)N

⟩ + eiφN−k∣∣D(N−k)

N

⟩]. (57)

Due to the strict symmetry constraint on the state, we obtainsmaller maximal values of macroscopicity for given geometricentanglement (dashed lines in Fig. 4) than for general states.

IV. STATISTICS OF MACROSCOPIC AND GEOMETRICENTANGLEMENT

Having established the relationship between geometricand macroscopic entanglement, we proceed to numericalinvestigations of pure states in different ensembles.

A. Random physical states

1. State generation

As a first ensemble of pure states, we consider randomphysical states, introduced in Refs. [46,47] and sketched inFig. 5(a). To generate a random physical state |�k〉, we assumethat the qubits are aligned in spin-chain configuration and

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FIG. 4. (Upper panel) Maximal value of normalized macroscop-icity as a function of the upper bound on EG, − log2 η, evaluatedusing the argument of Sec. III B, for general states (solid lines) andsymmetric states (dashed lines), for N = 3,6,10,20,30 (blue, red,black, green, and orange, respectively). The horizontal dotted lineindicates the limiting value 1/

√3 for symmetric states (Sec. IV C).

For N = 3, the bounds for general and for symmetric states coincide.Note that the geometric entanglement EG does not surpass N − 1.(Lower panel) Same relationship as above, but for the ad hoc–normalized EG/(N − 1). While the decay of M with EG becomesweaker with larger N on an absolute scale (upper panel), this is dueto the N -dependent range of EG.

that only pairwise interactions take place. We apply k timesa random two-particle unitary onto a randomly chosen pairof two neighboring qubits. That is, for k < N − 1, the stateremains at least 1-separable (the first qubit in the chain hasnever directly or indirectly interacted with the last one), whilewe obtain Haar-random states in the limit k � N , which wewill discuss separately in Sec. IV B below. To obtain a betterintuition for this ensemble, we compare random physical statesto random linear chains |�k〉, which are generated by applyinga binary unitary between the first k � N − 1 pairs of qubits,starting from a separable state [Fig. 5(b)]. As the authorsof [46,47] argue, the ensemble of random physical states canbe regarded as typical for physical systems that obey somelocality structure. A variant of random physical states forwhich the binary interactions are chosen to be not necessarilybetween adjacent neighbors but between any two randomlychosen qubits does not exhibit qualitative differences to thelocality-preserving model here.

FIG. 5. (a) Random physical states. We apply k times a randomunitary between two randomly chosen adjacent sites (closed boundaryconditions). For k � N , we have a fully connected system with highprobability, i.e., the state is typically 0 separable; for k → ∞, wereach the limit of Haar-uniform states. (b) Random linear chain. Weapply random unitary binary gates between the first k � N − 1 pairsof adjacent qubits. For k = N − 1, we have a fully connected system.Geometric entanglement (c) and (d) and normalized macroscopicity(e) and (f) for random physical states and random linear chains,for k random two-body unitaries, respectively. In (d), the geometricentanglement coincides for all numbers of qubits N . To comparedifferent system sizes, we plot the entanglement and the normalizedmacroscopicity as a function of the N -base logarithm of k in (c) and(e), and as a function of the normalized number of applied gates in (d)and (f). Sample size is 200; error bars show one standard deviation.

2. Numerical results

Geometric and macroscopic entanglement for randomphysical states are shown in Figs. 5(c) and 5(e) as a functionof the number of applied binary gates k, which can beconfronted to the behavior of random linear chains [Figs. 5(d)and 5(f)]. Between k ≈ N and k ≈ N2, we observe a steepincrease in the geometric entanglement of random physicalstates: For this range of numbers of binary interactions k,the state typically becomes fully inseparable. For k ≈ N3,we observe a saturation of both macroscopicity and geometricentanglement. While geometric entanglement increases mono-tonically with the number of applied gates, macroscopicitydevelops a peaked structure for N � 6. Consequently, thetrajectories of random physical states as a function of k in

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FIG. 6. Average trajectories of random physical states |�k〉 in theEG-M plane for N = 4,6,13,20. The solid lines start at k = 1 andproceed to k = N 3 (blue disks).

the (M,EG) plane (Fig. 6) proceed from low geometric andmacroscopic entanglement (k = 1) over a maximum to theasymptotic value with large geometric and low macroscopicentanglement. The maximum value of macroscopicity isreached for N � k � N2: In this range, the states can veryprobably not be decomposed into separable components, whileit remains moderately complex by construction—these are thevery requirements for high macroscopicity. Random linearchains feature a linear increase of geometric entanglementwith the number of applied gates, for which the curves forall particle numbers coincide [Fig. 5(d)], and a monotonicincrease of macroscopicity for k, peaking at lower and lowervalues as we increase the number of particles [Fig. 5(f)].

Macroscopicity as a function of the particle number N

is plotted in Fig. 7(a). The maximum value in randomphysical states decreases with increasing N (red diamonds),albeit slower than the saturated value of the macroscopicity(k = N3, black circles). The former also remains slightlylower than the macroscopicity reached for a saturated randomlinear chain (i.e., after k = N − 1 binary gates)—choosingthe interacting qubits randomly is disadvantageous for large

FIG. 7. (a) Normalized macroscopicity M after one random gate(k = 1, green squares) and after k = N3 random gates (black disks);maximal attained macroscopicity for random physical states (reddiamonds) and normalized macroscopicity for a binary chain withk = N − 1 random interactions (blue triangles), as a function of thenumber of qubits N . (b) Scaled macroscopicity NM ≈ M for thesame ensembles of states as in (a). Error bars show one standarddeviation.

macroscopicity, which gives an advantage to saturated linearchains. Complexity is adverse to macroscopicity: For N � 15,one randomly chosen binary interaction onto two qubits in aninitially separable state (k = 1, green squares) results in alarger macroscopicity than the limiting case k → N3.

The different types of decay raise the question of whether,albeit the fraction of particles participating in macroscopicsuperpositions decreases, the absolute number may in fact beconstant or increase. In Fig. 7(b), we plot the absolute size ofthe macroscopic component NM ≈

√M—the approxima-

tion is justified for N � 1—, which resolves the qualitativedifferences between the ensembles: The absolute number ofparticles participating in a GHZ-like state remains constantfor states into which exactly one random gate has been applied(green squares); it decreases for k = N3 (black circles), butit increases with N for the maximally achieved value inrandom physical states and for saturated random linear chains(k = N − 1).

In conclusion, starting from a separable state and applyingrandom binary gates, we first explore the region in whichgeometric and macroscopic entanglement are synonymous(Sec. III A), such that both quantities initially grow withk. When the spin chain is fully inseparable, additionalinteractions contribute to larger geometric entanglement, butsimultaneously destroy its macroscopicity ensuring that thelatter decreases (Sec. III B). Even though the absolute size ofthe macroscopic component increases with N [Fig. 7(b)], thefraction of particles participating in a macroscopic superposi-tion does not. To come back to our classical analogy (Fig. 1),just like there are no concerted forces that spontaneouslypush all gas particles to one side of the box, randomevolutions are unlikely to force all spins into a macroscopicsuperposition.

B. Haar-random states

1. State generation

In the limit k → ∞, random physical states converge toHaar-random states, i.e., the ensemble of pure quantum statesthat are uniformly distributed on the unit sphere in Hilbertspace [48]. Instead of applying many binary gates, one canconstruct Haar-random states by randomly generating the realand imaginary part of each state coefficient cj1,...,j2N

followinga zero-mean unit-variance normal distribution; the resultingunnormalized vector �c is then normalized in a second step.This procedure yields a “chaotic” ensemble [49] that remainsinvariant under local basis rotations and re-partitioning of theHilbert space into subsystems [50]. Random states also resultfrom the application of a Haar-random unitary matrix on anyconstant pure state [51].

2. Macroscopicity is rare in Haar-random states

Random states feature the concentration of measure phe-nomenon, i.e., most states on the high-dimensional Blochsphere lie close to the equator [52,53]. Given a Lipschitz-continuous function f (|�〉), the function values remain closeto the average value 〈f 〉 for the vast majority of states, reflected

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by the probability for a deviation larger than ε [54],

P [|f (|�〉) − 〈f 〉| > ε] � 4e− (n+1)ε2

24π2η2 , (58)

where η is the Lipschitz constant. The variance definedin Eq. (3) is Lipschitz continuous, since it is a sum ofbounded operators. The maximization procedure in Eq. (7)does not affect Lipschitz continuity, hence the unnormal-ized macroscopicity M remains Lipschitz continuous. Thegeometric measure of entanglement inherits Lipschitz con-tinuity from the distancelike measure it is based on [52].Hence, most Haar-random states are very similar, both whencharacterized by their geometric entanglement and by theirunnormalized macroscopicity M. Applying the normalizationto the macroscopicity via the square root in Eq. (7), welose Lipschitz continuity; strictly speaking, thus, M doesnot need to exhibit typical behavior. When restricted to arange with fixed lower bound, M � εM > 0, M is againLipschitz continuous, such that, eventually, most uniformlyrandomly chosen quantum states will not only share the sameunnormalized macroscopicityM but also the same normalizedmacroscopicity M.

Using random matrix theory [55], one can estimate thetypical magnitudes of the elements of the variance-covariancematrix (17) [49]. In the limit N → ∞, the VCM approachesthe unit matrix and, as the largest eigenvalues converge tounity, by the upper bound (21), the normalized macroscopicityvanishes. Qualitatively, this result also follows from thefollowing complementary argument: Random states that arechosen according to the Haar measure possess large geometricentanglement [20,21]. With probability greater than 1 − e−N2

,

we have for N � 11 [20],

EG(|�random〉) � N − 2 log2 N − 3. (59)

The upper bound on macroscopicity in Sec. III B then impliesthat the typical macroscopicity is necessarily small: Stronglygeometrically entangled states cannot be macroscopic. Thelower bound on EG (59) and the upper bound on macro-scopicity as a function of geometric entanglement within theconstruction of Sec. III B are, however, not sufficiently tightto make this argument quantitative.

3. Numerical results

The expected behavior is reproduced by our numerical data,shown in Fig. 8. In agreement with the previous argument,the geometric entanglement increases as a function of thenumber of qubits N (c), while the normalized macroscop-icity decays (b). This decay is approximately exponential[logarithmic inset of Fig. 8(b)], and we can safely state thateven the unnormalized macroscopicity M [Eq. (4)], whichreflects the absolute size of the macroscopic component,decreases. The variances of both normalized macroscopicityand geometric entanglement decrease as well. The histogramin Fig. 8(d) shows the distribution of states for N = 4 in the(M,EG) plane, together with the bounds in the regime EG � 1and the conjectured bounds in the realm EG � 1.

In conclusion, both the relative and the absolute size ofthe largest macroscopic superposition in Haar-random statesdecreases with N . As a consequence, Haar-random states arevery geometrically entangled and feature little macroscopicity.

FIG. 8. Macroscopicity of Haar-random states. Error bars show one standard deviation. (a) Average macroscopicity and geometricentanglement for N = 3, . . . ,23; no error bars shown. (b) Macroscopicity as a function of the number of qubits. (Inset) Logarithmic plot withfit by an exponential decay. (c) Geometric entanglement. (Blue solid line) Maximally possible value of geometric entanglement EG = N − 1.(Dashed black line) Lower bound on geometric entanglement of random states [Eq. (59)]. Error bars are not visible. Sample sizes (a)–(c)N = 3 − 10,105; N = 11 − 15,104; N = 16 − 23,103. (d) Histogram for 105 Haar-random states of N = 4 qubits, together with the upperbound of Sec. III B (red solid line, EG � 1) and the state |�〉 parametrized as in Fig. 2 (red solid line and black dashed line, EG � 1).

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C. Random symmetric states

Colloquially speaking, Haar-random states are extremelycomplex and do not allow any efficient description [56]. Astate with large macroscopicity, on the other hand, can beapproximated by a superposition of eigenstates of the totalspin operator [Eq. (53)], and thereby permits an efficientdescription. Hence, complexity and macroscopicity are mu-tually exclusive properties, and we cannot expect to encountermacroscopic superpositions in structureless ensembles.

On the other hand, ensembles of random pure states thatare less complex may feature higher values of macroscopicity.In particular, permutationally symmetric states constitutean ensemble of states with rather low geometric entangle-ment [39,43,57,58]:

EG(|�sym〉) � log2(N + 1), (60)

due to the vastly reduced dimensionality N + 1 of the spaceof symmetric N -qubit states in contrast to the full Hilbertspace of size 2N . We choose the following ensemble ofsymmetric states: In the Dicke-state representation [Eq. (38)],the coefficients cj are chosen to be normally distributedrandom variables (with normal real and imaginary parts), andthe resulting states are normalized. Following this prescription,the ensemble is invariant under permutation-symmetric localunitary operations.

The very different behavior of geometric entanglement forHaar-random and random symmetric states is evident compar-ing Figs. 8(c) and 9(c). Consistent with their low geometric

entanglement, symmetric states feature exceptionally high androbust macroscopicity.

Expectation values of observables read

〈�s |σk ⊗ σl|�s〉 =N∑

p,q=0

c∗pcq

⟨D

(p)N

∣∣σk ⊗ σl

∣∣D(q)N

⟩. (61)

Since the cp are chosen randomly and independently, onlythe summands with p = q will contribute to the averagein the limit of many qubits N → ∞. The only nontrivialexpectation values that do not vanish on average are thetwo-qubit correlations along the same axis,

〈�s |σk ⊗ σk|�s〉 = 13 . (62)

Consequently, the matrix Vsym in Eq. (32) converges to

Vsym =(

1 + N − 1

3

)1, (63)

with obvious eigenvalues, and, in the limit N → ∞, wetherefore expect that the macroscopicity approaches

M(|�s〉) → 1√3. (64)

For finite N , off-diagonal nonvanishing correlations maycontribute further to the fluctuations, which is why the averagemacroscopicity converges to 1/

√3 from above. This behavior

is confirmed empirically in Figs. 9(a) and 9(b), where theaverage value of macroscopicity for symmetric states is plotted

FIG. 9. Macroscopicity of random symmetric states for a sample of 3000 random states. Error bars show one standard deviation.(a) Normalized macroscopicity against geometric entanglement, for N = 2, . . . ,128. (b) Average normalized macroscopicity as a function ofthe number of qubits N , the dashed black line shows the limiting value 1/

√3. (c) Average geometric entanglement as a function of N , the

solid line shows the theoretical maximum log2(N + 1) [Eq. (60)]. (d) Average normalized largest and smallest eigenvalues λ/(1 + (N − 1)/3)of the matrix Vsym for randomly chosen symmetric states. The largest eigenvalue is directly related to the macroscopicity via Eq. (33). Thesmallest eigenvalue solves the minimization problem that consists in finding the additive observable with the weakest fluctuations. For largeN , the local spin orientation is rather irrelevant for experiencing large fluctuations, as long as all local spin measurements are performed alongthe same axis.

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against the average geometric entanglement (a) and the numberof qubits (b).

Moreover, not only does the macroscopicity converge to afinite value, it is also very robust with respect to misalignmentof spin orientations: Since the smallest and largest eigenvaluesof Vsym [Eq. (32)] converge to the same value 1 + (N − 1)/3[Fig. 9(d)], the spin orientation becomes irrelevant in the limitN → ∞: Almost every additive observable for which thelocal spin orientations are all identical features macroscopicfluctuations on a random symmetric state. The equality oflocal spin orientations is crucial here: If these orientationswere chosen randomly and independently, the expectationvalue would hardly fluctuate, since most eigenvalues of thefull variance-covariance matrix V Eq. (17) are typically small.

V. CONCLUSIONS AND OUTLOOK

Many different approaches to entanglement eventuallyturn out to be equivalent, motivating the powerful conceptsof entanglement monotone and entanglement measure [59].Our results emphasize that macroscopic entanglement, asquantified by Eq. (7), should never be treated as a synonymfor a measure of entanglement: In particular, there are entan-gled states that feature vanishing macroscopic entanglement(Sec. III A 1).

Random physical states reflect this intricate relationship bytheir trajectory in the (EG,M) plane (Sec. IV A), converging toHaar-random states, which feature large geometric and smallmacroscopic entanglement. The typical size of macroscopicsuperpositions of random physical states grows, but notas fast as the system size—consequently, the normalizedmeasure of macroscopicity converges to 0 in the limit N →∞. Symmetric states are naturally much less geometricallyentangled and much more macroscopic, and it remains to bestudied whether there are ensembles beyond symmetric statesfor which the actual spin orientations in the definition of theadditive observables [Eq. (1)] are irrelevant. Such ensembleswould be experimentally valuable due to their robustness.

A recent study [22] put forward a bound on the quan-tum Fisher information in terms of geometric entanglement[Eq. (47)]. The quantum Fisher information translates es-sentially to the unnormalized macroscopicity in our context.Our study corroborates further the irrelevance of geometricentanglement for the quantum Fisher information, sincelarge geometric entanglement implies small quantum Fisherinformation.

A general bound on macroscopicity as a function ofgeometric entanglement that features the decrease of

macrocopicity with increasing geometric entanglement be-yond unity seems hard to obtain, since both quantities aredefined via a maximization procedure. We believe neverthelessthat our bound in Sec. III B can be improved considerably, andthat a proof for the extremality of |�(θ,ε)〉 can be found. Wedid not find any relationship between the closest separablestate |φ1,φ2, . . . ,φN 〉 and the maximizing spin orientation{�α1, . . . ,�αN }; such deeper connection would be valuable.

Control schemes that optimize multipartite entanglementimplicitly exploit the typicality of entangled states within theensemble of pure states [60]. Our results suggest that controlstrategies that aim at a macroscopically entangled target statewill not only be affected by decoherence, but the unitaryevolution also needs to be tailored in a much more precise way:While the manifold of states that feature high geometric entan-glement is very large, this is not true for macroscopic states.

Finally coming back to our proposed analogy (Fig. 1),our results suggest that macroscopically entangled states playthe role of four-leaf clover: They do not appear sponta-neously after some random process, but only as the resultof some meticulously designed artificial evolution, such asin a quantum computer [61]. In other words, while thedynamical decoherence remains a pillar for the understandingof the emergence of classicality, already a purely statisticalexplanation satisfactorily explains the absence of macroscopicsuperpositions. On the other hand, the prevalence of macro-scopicity in random symmetric states (Sec. IV C) underlinesthe importance of the choice of the ensemble for (un)typicality.Hence, further investigations of other ensembles of purequantum states, such as canonical thermal pure states [62]and random matrix product states [63,64], will eventuallyrigorously confirm or dismiss the analogy.

ACKNOWLEDGMENTS

C.-Y.P., M.K., and H.J. were supported by the NationalResearch Foundation of Korea (NRF) grant funded by theKorea Government (MSIP) (Grant No. 2010-0018295). M.C.Tand K.M. acknowledge financial support from the DanishCouncil for Independent Research and the Villum Foundation.M.C.T. was financially supported by the bilateral DAAD-NRF scientist exchange programme. C.-Y.P. thanks JinhyoungLee and Seokwon Yoo for granting access to the Alicecluster system at the Quantum Information Group, HanyangUniversity. The authors thank Christian Kraglund Andersen,Ralf Blattmann, Eliska Greplova, Qing Xu, and Jinglei Zhangfor valuable comments on the manuscript.

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